In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale.

Due to the principle of octave equivalence, scales are generally considered to span a single octave, with higher or lower octaves simply repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval) between two successive notes of the scale.[2] However, there is no need for scale steps to be equal within any scale and, particularly as demonstrated by microtonal music, there is no limit to how many notes can be injected within any given musical interval.

A measure of the width of each scale step provides a method to classify scales. For instance, in a chromatic scale each scale step represents a semitone interval, while a major scale is defined by the interval pattern T–T–S–T–T–T–S, where T stands for whole tone (an interval spanning two semitones), and S stands for semitone. Based on their interval patterns, scales are put into categories including diatonic, chromatic, major, minor, and others.

A specific scale is defined by its characteristic interval pattern and by a special note, known as its first degree (or tonic). The tonic of a scale is the note selected as the beginning of the octave, and therefore as the beginning of the adopted interval pattern. Typically, the name of the scale specifies both its tonic and its interval pattern. For example, C major indicates a major scale with a C tonic.

Scales are typically listed from low to high pitch. Most scales are octave-repeating, meaning their pattern of notes is the same in every octave (the Bohlen–Pierce scale is one exception). An octave-repeating scale can be represented as a circular arrangement of pitch classes, ordered by increasing (or decreasing) pitch class. For instance, the increasing C major scale is C–D–E–F–G–A–B–[C], with the bracket indicating that the last note is an octave higher than the first note, and the decreasing C major scale is C–B–A–G–F–E–D–[C], with the bracket indicating an octave lower than the first note in the scale.

The distance between two successive notes in a scale is called a scale step.

The notes of a scale are numbered by their steps from the root of the scale. For example, in a C major scale the first note is C, the second D, the third E and so on. Two notes can also be numbered in relation to each other: C and E create an interval of a third (in this case a major third); D and F also create a third (in this case a minor third).

A single scale can be manifested at many different pitch levels. For example, a C major scale can be started at C4 (middle C; see scientific pitch notation) and ascending an octave to C5; or it could be started at C6, ascending an octave to C7. As long as all the notes can be played, the octave they take on can be altered.

Scales may also be described by their constituent intervals, such as being hemitonic, cohemitonic, or having imperfections[3]. Many music theorists concur that the constituent intervals of a scale have a large role in the cognitive perception of its sonority, or tonal character.

"The number of the notes that make up a scale as well as the quality of the intervals between successive notes of the scale help to give the music of a culture area its peculiar sound quality."[4] "The pitch distances or intervals among the notes of a scale tell us more about the sound of the music than does the mere number of tones."[5]

Scales can also be described by their distribution patterns and possibilities for notation. For example, a heliotonic scale is one that can be notated with one note head on each line and space, using only single and double alterations (flats, double-flats, sharps, and double-sharps). Thus all heliotonic scales are heptatonic, but not all heptatonic scales are heliotonic. Since heliotonia is a metric of a scale's tone distribution pattern, is related to evenness, coherence, spectra variation, and the Myhill Property.

The notes of a scale form intervals with each of the other notes of the chord in combination. A 5-note scale has 10 of these harmonic intervals, a 6-note scale has 15, a 7-note scale has 21, an 8-note scale has 28.[7] Though the scale is not a chord, and might never be heard more than one note at a time, still the absence, presence, and placement of certain key intervals plays a large part in the sound of the scale, the natural movement of melody within the scale, and the selection of chords taken naturally from the scale.[7]

A musical scale that contains tritones is called tritonic (though the expression is also used for any scale with just three notes per octave, whether or not it includes a tritone), and one without tritones is atritonic. A scale or chord that contains semitones is called hemitonic, and without semitones is anhemitonic.

Scales can be abstracted from performance or composition. They are also often used precompositionally to guide or limit a composition. Explicit instruction in scales has been part of compositional training for many centuries. One or more scales may be used in a composition, such as in Claude Debussy's L'Isle Joyeuse.[8] To the right, the first scale is a whole-tone scale, while the second and third scales are diatonic scales. All three are used in the opening pages of Debussy's piece.

Scales in traditional Western music generally consist of seven notes and repeat at the octave. Notes in the commonly used scales (see just below) are separated by whole and half stepintervals of tones and semitones. The harmonic minor scale includes a three-semitone step; the anhemitonic pentatonic includes two of those and no semitones.

These scales are used in all of their transpositions. The music of this period introduces modulation, which involves systematic changes from one scale to another. Modulation occurs in relatively conventionalized ways. For example, major-mode pieces typically begin in a "tonic" diatonic scale and modulate to the "dominant" scale a fifth above.

Scales such as the pentatonic scale may be considered gapped relative to the diatonic scale. An auxiliary scale is a scale other than the primary or original scale. See: modulation (music) and Auxiliary diminished scale.

In many musical circumstances, a specific note of the scale is chosen as the tonic—the central and most stable note of the scale. In Western tonal music, simple songs or pieces typically start and end on the tonic note. Relative to a choice of a certain tonic, the notes of a scale are often labeled with numbers recording how many scale steps above the tonic they are. For example, the notes of the C major scale (C, D, E, F, G, A, B) can be labeled {1, 2, 3, 4, 5, 6, 7}, reflecting the choice of C as tonic. The expression scale degree refers to these numerical labels. Such labeling requires the choice of a "first" note; hence scale-degree labels are not intrinsic to the scale itself, but rather to its modes. For example, if we choose A as tonic, then we can label the notes of the C major scale using A = 1, B = 2, C = 3, and so on. When we do so, we create a new scale called the A minor scale. See the musical note article for how the notes are customarily named in different countries.

The scale degrees of a heptatonic (7-note) scale can also be named using the terms tonic, supertonic, mediant, subdominant, dominant, submediant, subtonic. If the subtonic is a semitone away from the tonic, then it is usually called the leading-tone (or leading-note); otherwise the leading-tone refers to the raised subtonic. Also commonly used is the (movable do) solfège naming convention in which each scale degree is denoted by a syllable. In the major scale, the solfège syllables are: do, re, mi, fa, so (or sol), la, ti (or si), do (or ut).

In naming the notes of a scale, it is customary that each scale degree be assigned its own letter name: for example, the A major scale is written A–B–C♯–D–E–F♯–G♯ rather than A–B–D♭–D–E–E–G♯. However, it is impossible to do this in scales that contain more than seven notes, at least in the English-language nomenclature system.[citation needed]

Scales may also be identified by using a binary system of twelve zeros or ones to represent each of the twelve notes of a chromatic scale. It is assumed that the scale is tuned using 12-tone equal temperament (so that, for instance, C♯ is the same as D♭), and that the tonic is in the leftmost position. For example, the binary number 101011010101, equivalent to the decimal number 2773, would represent any major scale (such as C–D–E–F–G–A–B). This system includes scales from 100000000000 (2048) to 111111111111 (4095), providing a total of 2048 possible species, but only 351 unique scales containing from 1 to 12 notes.[9]

A more mathematically flexible schema for binary representation, espoused by theorist Ian Ring in his study of musical scales,[10] has bits representing ascending tones, with the smaller bits representing lower pitches and higher bits representing higher pitches. This causes the bits to appear reversed compared with Duncan's system. Under this schema, a major scale is 101010110101 = (2741), a single root tone is 1, and a chromatic scale is still 111111111111 (4095). All the tone possibilities that begin with a root tone, and are thus scale-like, are odd-numbered.

Scales may also be shown as semitones from the tonic. For instance, 0 2 4 5 7 9 11 denotes any major scale such as C–D–E–F–G–A–B, in which the first degree is, obviously, 0 semitones from the tonic (and therefore coincides with it), the second is 2 semitones from the tonic, the third is 4 semitones from the tonic, and so on. Again, this implies that the notes are drawn from a chromatic scale tuned with 12-tone equal temperament. For some fretted string instruments, such as the guitar and the bass guitar, scales can be notated in tabulature, an approach which indicates the fret number and string upon which each scale degree is played.

Composers transform musical patterns by moving every note in the pattern by a constant number of scale steps: thus, in the C major scale, the pattern C–D–E might be shifted up, or transposed, a single scale step to become D–E–F. This process is called "scalar transposition" or "shifting to a new key" and can often be found in musical sequences and patterns. (It is D-E-F# in Chromatic transposition). Since the steps of a scale can have various sizes, this process introduces subtle melodic and harmonic variation into the music. In Western tonal music, the simplest and most common type of modulation (or changing keys) is to shift from one major key to another key built on the first key's fifth (or dominant) scale degree. In the key of C major, this would involve moving to the key of G major (which uses an F#). Composers also often modulate to other related keys. In some Romantic music era pieces and contemporary music, composers modulate to "remote keys" that are not related to or close to the tonic. An example of a remote modulation would be taking a song that begins in C major and modulating (changing keys) to F# major.

Through the introduction of blue notes, jazz and blues employ scale intervals smaller than a semitone. The blue note is an interval that is technically neither major nor minor but "in the middle", giving it a characteristic flavour. A regular piano cannot play blue notes, but with electric guitar, saxophone, trombone and trumpet, performers can "bend" notes a fraction of a tone sharp or flat to create blue notes For instance, in the key of E, the blue note would be either a note between G and G♯ or a note moving between both.

In blues, a pentatonic scale is often used. In jazz, many different modes and scales are used, often within the same piece of music. Chromatic scales are common, especially in modern jazz.

In Western music, scale notes are often separated by Equal temperament tones or semitones, creating 12 intervals per octave each separating two tones where the higher tone has a oscillation frequency of a fixed ratio (by a factor equal to the twelfth root of two, or approximately 1.059463) higher than the frequency of the lower one. A scale uses a subset consisting typically of 7 of these 12 as scale steps. Other traditional tempered scale systems divide the octave to equal intervals by numbers other than 12. Examples are 53 in the Middle East, and 72 in southern India.[citation needed]

Many other musical traditions use scales that include other intervals. These scales originate within the derivation of the harmonic series. Musical intervals are complementary values of the harmonicovertones series.[11] Many musical scales in the world are based on this system, except most of the musical scales from Indonesia and the Indochina Peninsulae, which are based on inharmonic resonance of the dominant metalophone and xylophone instruments.

Some scales use a different number of pitches. A common scale in Eastern music is the pentatonic scale, which consists of five notes that span an octave.

Some scales span part of an octave; Several such short scales are typically combined to form a scale spanning a full octave or more, and usually called with a third name of its own. The Middle Eastern music has around a dozen such basic short scales that are combined to form hundreds of full-octave spanning scales. Among these middle eastern scales Hejaz scale has one scale step spanning 14 intervals (of the middle eastern type found 53 in an octave) roughly similar to 3 semitones (of the western type found 12 in an octave), while Saba scale, another of these middle eastern scales, has 3 consecutive scale steps within 14 commas, i.e. separated by roughly one western semitone either side of the middle tone.